Cognitive Sciences Stack Exchange is a question and answer site for practitioners, researchers, and students in cognitive science, psychology, neuroscience, and psychiatry. It's 100% free, no registration required.

3 Answers
3

Savant syndrome is poorly understood. No widely accepted cognitive theory explains the combination of talent and deficit found in savants.
It has been suggested that individuals with autism are biased towards detail-focused processing and that this cognitive style predisposes both individuals with and without autism to savant talents.

Another hypothesis is that hyper-systemizing predisposes people to show talent, where hyper-systemizing is an extreme state in the empathizing–systemizing theory that classifies people based on their skills in empathizing with others versus systemizing facts about the external world, and that the attention to detail shown by many savants is a consequence of enhanced perception or sensory hypersensitivity in individuals with autism.

It has also been suggested that savants operate by directly accessing low-level, less-processed information that exists in all human brains but is normally not available to conscious awareness.

How one performs quick mental calculations through tricks and shortcuts can easily be looked up on the internet. How some Savant's do it cannot because we don't know. There is some argument that some do those same tricks but others seem not to do so.

Consider that a substantial amount of computational power goes into immediately recognizing that your friend is deeply unhappy when you first see them but puts on a brave face for you and is avoiding talking about something deeply troubling them. You see that immediately while the typical savant sees nothing, or confusion. That same Savant can do amazing mathematical, or spatial computations with tremendous facility. So, perhaps the reason we don't understand it is because it's actually a different kind of a mind.

Savants apparently don't compute consciously; from what I've seen, they see numbers as shapes. When savants multiply numbers together, they see two shapes. The image starts to change and evolve, and a third shape emerges. That's the answer.

Humans can have synesthesia. It is a process in the brain which can associate numbers or names to certain shapes. For example, consider the two shapes:

Tests like this demonstrate that people do not attach sounds to visual shapes arbitrarily. Which shape would you call "Bouba" and which "Kiki"?
I would call the one on the left "kiki" and the one on the right "Bouba".

You can see this interesting video on the wonderful savant Daniel Tammet. Start from 32:16 if you want to see how a neurology team administered various tests to him to make sure he's not using some techniques. See also how he uses "Kiki-Bouba"-like shapes to do complex computations at 40:35.